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# Show that the points $(-2,3,5),\:(1,2,3)\:and\:(7,0,-1)$ are collinear.

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• To prove three points $A,B,C$ are collinear, we should prove sum of the distances between any two pair points is equal to that of the third pair.
• Distance between the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is given by $\sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Let $A(-2,3,5),\:B(1,2,3)\:and\:C(7,0,-1)$ be the given three points.
We know that the distance between the points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is given by
$\sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
Distance $\overline {AB}=\sqrt {(1+2)^2+(2-3)^2+(3-5)^2}=\sqrt{9+1+4}=\sqrt {14}$
Distance $\overline {BC}=\sqrt {(7-1)^2+(0-2)^2+(-1-3)^2}=\sqrt{36+4+16}=\sqrt {56}=2\sqrt {14}$
Distance $\overline {AC}=\sqrt {(7+2)^2+(0-3)^2+(-1-5)^2}=\sqrt{81+9+36}=\sqrt {126}=3\sqrt {14}$
$\Rightarrow\:\overline {AB}+\overline {BC}=\overline {AC}$
$\Rightarrow\:$ the three points $A,B,C$ are collinear.
answered Apr 17, 2014

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